It seems to me he's just used the Rank Theorem for manifolds (which I state for your convenience) to give a canonical "flattened out" form to the inclusion map:
Given (1) smooth manifolds M, N of dimensions m, n resp.
(2) a smooth map f : M --> N of constant rank k
(3) any point p in M
Then, there exist charts (G, g) and (H, h) "centred on" p and f(p) resp., f(G) contained in H such that the "coordinate representative" F of f w.r.t. the charts, acts as
(x_1, ..., x_k, ..., x_m) --> (x_1, ..., x_k, 0, ..., 0).
[i.e. F : g^(-1)(G) \subset IR^m --> h(H) \subset IR^n, and F = hfg^(-1).]
In your case, f = i, k = m, while U = g^(-1)(G) and H have been quietly chosen as per the theorem. U is thus not just any parameter domain.
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